Sunderland 0 Liverpool 2: For Luis Suarez and Andy Carroll there was always going to be some degree of malice in Sunderland.

For Suarez, there was a reminder from the home side's Ghanaian contingent that he happens to be "the most hated man" in the African country, thanks to the Uruguayan's cynical goal-denying handball in the World Cup quarter-final in Johannesburg last July. For Carroll, there was that special Wearside welcome they reserve for those who have been closely associated with Newcastle United.

As it happened, Liverpool's £58m strike force took their share of stick from the natives in the capacity 47,207 crowd as they foraged in tandem from the off for the first time. Carroll, at £35m the most expensive British player in history, endured taunts of "What a waste of money" as he made way for David N'Gog with 18 minutes remaining, having drawn a blank on his first start for his new club. At that point, the ire of the Sunderland supporters was directed mainly at the match officials – referee Kevin Friend having been persuaded by one of his assistants, Billy Smallwood, to award the penalty kick with which Dirk Kuyt gave the visitors a 32nd-minute lead.

The sense of injustice was certainly valid, television replays having shown that Jay Spearing was two yards outside the Sunderland box when John Mensah's clumsy challenge set him on a dive that finished well beyond the 18-yard line. But then, with one flash of brilliance 13 minutes from time, the grievance was rendered irrelevant.

It came from Suarez. Drifting wide to the right to meet a throw-in from Kuyt, he slipped past Lee Cattermole and rifled a right-foot shot in between Simon Mignolet, the Sunderland goalkeeper, and his near post from the tightest of angles. That is what you get for £22.8m: another drop of the kind of close-quarters magic that bamboozled Manchester United a fortnight ago.

King Kenny had his sixth win in the bag in 10 Premier League games since his return to the managerial throne at Anfield. For Steve Bruce and Sunderland, it was a sixth defeat in seven. They started strongly but finished with 10 men, Mensah – one of their Ghanaian World Cup quarter-final veterans – being shown a red card for holding back Suarez as the South American attempted to race clear in the 82nd minute.

"Luis and Andy showed a lot of encouraging signs," Dalglish said. "Andy showed great fitness to play 90 minutes on Thursday and another 70 today. Luis chipped in with a fantastic goal. They complemented each other well."

Carroll made his 6ft 3in presence felt in patches, not least in the fifth minute, when he got his head to a corner from Raul Meireles and knocked the ball down for Kuyt to force a save from Mignolet. For much of the first-half, however, Liverpool were on the back foot.

Indeed, they were lucky to survive unscathed when Danny Welbeck combined superbly with Asamoah Gyan on the left and whipped in a teasing cross that Kieran Richardson failed to connect with by the merest whisker. Just as Sunderland were getting into their stride, though, they were forced to reorganise, Sulley Muntari and Richardson departing with injuries and Cattermole and Steed Malbranque arriving from the bench.

The biggest jolt, however, came in the 32nd minute, when Mensah failed to trap a long ball with his chest and was obliged to haul down Spearing. Despite Sunderland's protests over the penalty award – "It was a blatant error," Bruce maintained afterwards – Kuyt made no mistake with the kick and Liverpool were in the driving seat.

They might have been 2-0 up before half-time but Mignolet saved when Suarez was put in the clear by Miereles. It was all over, though, when Suarez foxed Mignolet, leaving Dalglish to fend questions about his long-term future. "I've read that I'm supposed to have asked for four years and been offered two," he said. "There have been no detailed discussions."